where \(\Delta _\mathcal {U}\)  is the semisimplicial object in \({\bf {Top}}_X\) 
associated with an open cover \(\mathcal {U}\)  of \(X\) . Next, one sees that
locally the groupoid of deformations of \(\mathcal {E}|_U\)  is equivalent to
the Deligne groupoid of \(\mathcal {E}nd(\mathcal {E})(U)\) ; since these
equivalences are compatible with restriction maps, one has an
equivalence of semicosimplicial groupoids. Finally, Deligne
groupoid commutes with homotopy limits of DGLA concentrated in
positive degree (see [1]}), so that
\( \operatorname{Def}_\mathcal {E}\simeq \mathop {\rm holim}_{U\in \Delta _\mathcal {U}} \operatorname{Del}_{\mathcal {E}nd(\mathcal {E})(U)}\simeq \operatorname{Del}_{\begin{array}{c}{\mathop {\rm holim}\mathcal {E}nd(\mathcal {E})(U)}\\{\scriptscriptstyle {U\in \Delta _\mathcal {U}\phantom{mmmmmi}}}\end{array}}.\) 
